AMA1006 Basic Statistics, AMA1104 Introductory Probability Assignment 1 Due1 at 17:00, February 22, 2016, Monday. Please submit your homework into the assignment box at oor 7 or 8, core T. The assignment box will be available for about one week ahead of the deadline, and with the lecturer's name printed on the front. The markers of the assignments are research students from department AMA. Please show your courtesy to them by boxing your nal solutions, like: ...therefore the desired probability is P (A) = 0.334 . Full marks: 100. 1. (10 marks) To test the quality of a shipment of crystal glasses, we selected 62 glasses at random. We found that two were scratched and chipped, three had only scratches, and ve were only chipped. We randomly select two glasses, X and Y , out of these 62 ones. Consider the following events: A : {X is chipped}, and B : {Y is scratched }. Determine the relative frequencies of A, B, A B, and A B. Are the events A and B independent? 2. (20 marks) a. If two balanced dice are rolled, what is the probability that the sum of the two numbers that appear is odd? b. If two balanced dice are rolled, what is the probability that the sum of the two numbers that appear is even? c. If two balanced dice are rolled, what is the probability that the difference between the two numbers that appear will be no less than 5? d. If three fair coins are tossed, what is the probability that all three faces will be the same? 3. (20 marks) How many different ways can one make a 5-digit number using digits 1, 2, , 9 (but not 0), such that a. each digit appears at most once? b. each digit appears at most twice? c. each digit may appear any times? d. each digit appears at most once and digits in the number appear in an ascending order (like 13456 but not 14365)? Hint: Think each digit as a card. In (a) you have 9 such cards but in (b) you have 18. Use multiplicative rule for (c) and combinations rule for (d). 4. (10 marks) A box contains 24 light bulbs, of which two are defective. If Tom selects 10 bulbs at random, without replacement, what is the probability that both defective bulbs will be selected? What is the probability that only one defective bulbs will be selected? 1 FAQ: \"What if I am sick on the day of deadline?\" ANS: you are encouraged to submit the assignment days before the deadline; in the worst case scenario you may take photos of your whole assignment and email it to the lecturer before the deadline and we will print it out, and submit it to the marker for you; otherwise you would need a medical certicate for us to accept your late submission. 1 5. (20 marks) In USA Today (Sept.5,1996) the results of a survey involving the use of sleep wear while travelling were listed as follows: Underwear Nightgown Nothing Pajamas T-shirt Other Male 0.220 0.002 0.160 0.102 0.046 0.084 Female 0.024 0.180 0.018 0.073 0.088 0.003 Total 0.244 0.182 0.178 0.175 0.134 0.087 a. What is the probability that a traveler is a male who sleeps in the nude? b. What is the probability that a traveler is female? c. Assuming the traveler is a female, what is the probability that he sleeps in pajamas? d. What is the probability that a traveler is male, given that he sleeps in Underwear or a T-shirt? 6. (10 marks) The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.17; the probability that it is a camper is 0.38; and the probability that it is a camper with Canadian license plates is 0.12. What is the probability that a. a camper entering the Luray Caverns has Canadian license plates? b. a vehicle with Canadian license plates entering the Luray Caverns is a camper? c. a vehicle entering the Luray Caverns is neither a Canadian plates holder, nor a camper? 7. (10 marks) In a certain region of the country it is known from the past experience that among the adults over 40 years of age, 1% have cancer. With the state-of-the-art diagnostic technology, if an adult has cancer, there is a probability of 0.88 that a medical test will give a positive response; whereas, if an adult does not have cancer, there is a probability of only 0.02 that the test will still give a positive response. Suppose Tom, an adult over 40, has just got a positive response in this test, what is the probability that he has cancer? 2